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Estimation theory is a branch of statistics and signal processing that deals with estimating the values of parameters based on measured/empirical data. The parameters describe the physical scenario or object that answers a question posed by the estimator. For example, it is desired to estimate the proportion of a population of voters who will vote for a particular candidate. That proportion is the unobservable parameter; the estimate is based on a small random sample of voters. Or, for example, in radar the goal is to estimate the location of objects (airplanes, boats, etc.) by analyzing the received echo and a possible question to be posed is "where are the airplanes?" To answer where the airplanes are, it is necessary to estimate the distance the airplanes are at from the radar station, which can provide an absolute location if the absolute location of the radar station is known. In estimation theory, it is assumed that the desired information is embedded into a noisy signal. Noise adds uncertainty and if there was no uncertainty then there would be no need for estimation.
Fields that use estimation theory There are numerous fields that require the use of estimation theory. Some of these fields include (but by no means limited to): The measured data is likely to be subject to noise or uncertainty and it is through statistical probability that optimal solutions are sought to extract as much information from the data. Estimation process The entire purpose of estimation theory is to arrive at an estimator, and preferably an implementable one that could actually be used. The estimator takes the measured data as input and produces an estimate of the parameters. It is also preferable to derive an estimator that exhibits optimality. An optimal estimator would indicate that all available information in the measured data has been extracted, for if there was unused information in the data then the estimator would not be optimal. These are the general steps to arrive at an estimator: After arriving at an estimator, real data might show that the model used to derive the estimator is incorrect, which may require repeating these steps to find a new estimator. A non-implementable or infeasible estimator may need to be scrapped and the process start anew. In summary, the estimator estimates the parameters of a physical model based on measured data. Basics To build a model, several statistical "ingredients" need to be known. These are needed to ensure the estimator has some mathematical tractability instead of being based on "good feel". The first is a set of statistical samples taken from a random vector (RV) of size . Put into a vector, . Secondly, we have the corresponding parameters , which need to be established with their probability density function (pdf) or probability mass function (pmf) . It is also possible for the parameters themselves to have a probability distribution (e.g., Bayesian statistics). It is then necessary to define the epistemic probability . After the model is formed, the goal is to estimate the parameters, commonly denoted , where the "hat" indicates the estimate. One common estimator is the minimum mean squared error (MMSE) estimator, which utilizes the error between the estimated parameters and the actual value of the parameters as the basis for optimality. This error term is then squared and minimized for the MMSE estimator. Estimators This list is some of the more common estimators used, and some topics related to them: Example: DC gain in white Gaussian noise Consider a received discrete signal, , of independent samples that consists of a DC gain with Additive white Gaussian noise with known variance (i.e., ). Since the variance is known then the only unknown parameter is . The model for the signal is then Two possible (of many) estimators are: Both of these estimators have a mean of , which can be shown through taking the expected value of each estimator and mathrmleft hat{A}_2 ight mathrmleft rac{1}{N} sum_{n=0}^{N-1} xn ight rac left sum_{n=0}^{N-1} mathrm{E}left xn ight ight rac left N A ight A At this point, these two estimators would appear to perform the same. However, the difference between them becomes apparent when comparing the variances. ight) = mathrm left( x0 ight) = sigma^2 and mathrm left( hat_2 ight) mathrm left( rac sum_^ xn ight) rac left sum_{n=0}^{N-1} mathrm{var} (xn) ight rac left N sigma^2 ight rac It would seem that the sample mean is a better estimator since, as , the variance goes to zero. Maximum likelihood Continuing the example using the maximum likelihood estimator, the probability density function (pdf) of the noise for one sample is ight) and the probability of becomes ( can be thought of a ) ight) By independence, the probability of becomes p(mathbf; A) prod_^ p(xn; A) rac expleft(- rac sum_^(xn - A)^2 ight) Taking the natural logarithm of the pdf ln p(mathbf; A) -N ln left(sigma sqrt ight) - rac sum_^(xn - A)^2 and the maximum likelihood estimator is Taking the first derivative of the log-likelihood function rac ln p(mathbf; A) rac left sum_{n=0}^{N-1}(xn - A) ight rac left sum_{n=0}^{N-1}xn - N A ight and setting it to zero 0 rac left sum_{n=0}^{N-1}xn - N A ight sum_^xn - N A This results in the maximum likelihood estimator hat = rac sum_^xn which is simply the sample mean. From this example, it was found that the sample mean is the maximum likelihood estimator for samples of AWGN with a fixed, unknown DC gain. Cramér-Rao lower bounds To find the Cramér-Rao lower bounds (CRLB) of the sample mean estimator, it is first necessary to find the Fisher information number mathcal(A) mathrm left( left rac{partial}{partial heta} ln p(mathbf{x}; A) ight^2 ight) -mathrm left rac{partial^2}{partial heta^2} ln p(mathbf{x}; A) ight and copying from above rac ln p(mathbf; A) rac left sum_{n=0}^{N-1}xn - N A ight Taking the second derivative rac ln p(mathbf; A) rac (- N) rac and finding the negative expected value is trivial since it is now a deterministic constant Finally, putting the Fisher information into mathrmleft( hat ight) geq rac results in mathrmleft( hat ight) geq rac Comparing this to the variance of the sample mean (determined previously) shows that the sample mean is equal to the Cramér-Rao lower bounds for all values of and . The sample mean is the minimum variance unbiased estimator (MVUE) in addition to being the maximum likelihood estimator. This example of DC gain + WGN is a recurring example in Kay's Fundamentals of Statistical Signal Processing. Books See also | ||||||||
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